# 4.5: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

1. Goodyear AZ is one of the fastest growing cities in the nation according to the census bureau. In 2012, the population was about 72,800. The city’s population grew by 3800 people from 2012 to 2013. If the growth keeps up in a linear fashion, create a population model for Goodyear. How many people will live there in 10 years? How many people will live there in 50 years? (U.S. Census, 2014)
1. Gilbert AZ is one of the fastest growing cities in the nation according to the census bureau. In 2012, the population was about 229,800. The city’s population grew by 9200 people from 2012 to 2013. If the growth keeps up in a linear fashion, create a population model for Gilbert. How many people will live there in 10 years? How many people will live there in 50 years? (U.S. Census, 2014)
1. Apache County AZ, is shrinking according to the census bureau. In 2012, the population was about 71,700. The county’s population decreased by 1147 people from 2012 to 2013. If the decline is linear, create a population model for Apache County. How many people will live there in 10 years? How many people will live there in 50 years? (Kiersz, 2015)
1. Cochise County AZ, is shrinking according to the census bureau. In 2012, the population was about 129,472. The county’s population decreased by 2600 people from 2012 to 2013. If the decline is linear, create a population model for Cochise County. How many people will live there in 10 years? How many people will live there in 50 years? (Kiersz, 2015)
1. Mohave County AZ, is shrinking according to the census bureau. In 2012, the population was about 203,030. The county’s population decreased by 1200 people from 2012 to 2013. If the decline is linear, create a population model for Mohave County. How many people will live there in 10 years? How many people will live there in 50 years? (Kiersz, 2015)
1. The 2012 Kia Sedona LX has one of the largest depreciation values of any car. Suppose a 2012 Kia Sedona sold for $24,900, and its value depreciates by$3400 per year. Assuming the depreciation is linear, find a model for the depreciation. How much is the car worth in five years? How much is the car worth in 10 years? When is it worth nothing? (Fuscaldo, n.d.)
1. The 2013 Chevy Impala has one of the largest depreciation values of any car. Suppose a 2013 Chevy Impala sold for $27,800 and its value depreciates by$3600 per year. Assuming the depreciation is linear, find a model for the depreciation. How much is the car worth in five years? How much is the car worth in 10 years? When is it worth nothing? (Fuscaldo, n.d.)
1. The 2013 Jaguar XJ AWD has one of the largest depreciation values of any car. Suppose a 2013 Jaguar XJ AWD sold for $74,500 and its value depreciates by$10,400 per year. Assuming the depreciation is linear, find a model for the depreciation. How much is the car worth in five years? How much is the car worth in 10 years? When is it worth nothing? (Fuscaldo, n.d.)
1. The 2012 Jeep Liberty Limited Sport 2WD has one of the largest depreciation values of any car. Suppose a 2012 Jeep Liberty Limited Sport 2WD sold for $23,400 and its value depreciates by$3,040 per year. Assuming the depreciation is linear, find a model for the depreciation. How much is the car worth in five years? How much is the car worth in 10 years? When is it worth nothing? (Fuscaldo, n.d.)
1. Suppose that in January, the maximum water depth at Lake Powell, Arizona was 528 feet. The water evaporates at an average rate of 1.2 feet per month. Find a model for the rate at which the water evaporates. If it does not rain at all, what will be the depth of Lake Powell in May and in September?
1. Suppose that the maximum water depth at Lake Tahoe, California in 2014 was 1644 feet. Because of the drought, the water level has been decreasing at an average rate of 6.2 feet per year. Find a model for the rate at which the water level decreases. If it there is no precipitation at all, what will be the depth of Lake Tahoe be in two years, and in five years?
2. Suppose the homes in Arizona have appreciated an average of 8% per year in the last five years. If the average home in a suburb sold for $225,000 in 2010, create a model for the home prices in the suburb. How much would this home be worth in 2015? 1. Suppose the homes in Massachusetts have appreciated an average of 13% per year in the last five years. If the average home in a suburb sold for$205,000 in 2010, create a model for the home prices in the suburb. How much would this home be worth in 2015?
1. Suppose the homes in Michigan have depreciated an average of 17% per year in the last five years. If the average home in a suburb sold for $215,000 in 2010, create a model for the home prices in the suburb. How much would this home be worth in 2015? 1. Suppose the homes in Nevada have depreciated an average of 15% per year in the last five years. If the average home in a suburb sold for$318,000 in 2010, create a model for the home prices in the suburb. How much would this home be worth in 2015?
1. The cost of a home in Flagstaff AZ was $89,000 in 1992. In 2007, the same home appraised for$349,000. Assuming the home value grew according to the exponential growth model, find the annual growth rate of this home over this 15-year period. If the growth continued at this rate, what would the home be worth in 2020?
1. The cost of a home in Bullhead City AZ was $109,000 in 1992. In 2007, the same home appraised for$352,000. Assuming the home value grew according to the exponential growth model, find the annual growth rate of this home over this 15-year period. If the growth continued at this rate, what would the home be worth in 2020?
1. The population of West Virginia is in decline. The population in 2014 was 1,850,326 and the population had decreased by 0.14% from 2010. How many people were living in West Virginia in 2010? Create a model for this population. If the decline continues at this rate, how many people will reside in West Virginia in 2020? (Wikipedia, n.d.)
1. Assume that the population of Arizona grew by 2.4% per year between the years 2000 to 2010. The number of Native American living in Arizona was 257,426 in 2010. How many Native Americans were living in Arizona in 2000? Create a model for this population. If the growth continues at this rate, how many Native Americans will reside in Arizona in 2020?
1. Assume the population of the U.S. grew by 0.96% per year between the years 2000 and 2010. The number of Hispanic Americans was 55,740,000 in 2010. How many Hispanic Americans were living in the U.S. in 2000? Create a model for this population. If the growth continues at this rate, how many Hispanic Americans will reside in The U.S. in 2020?
1. Assume the population of Michigan decreased by 0.6% per year between the years 2000 to 2010. The population of Michigan was 9,970,000 in 2010. How many people were living in Michigan in 2000? Create a model for this population. If the growth continues at this rate, how many people will reside in Michigan in 2020?
1. The doubling time of a population of aphids is 12 days. If there are initially 200 aphids, how many aphids will there be in 17 days?
1. The doubling time of a population of rabbits is six months. If there are initially 26 rabbits, how many rabbits will there be is 17 months?
1. The doubling time of a population of shrews is three months. If there are initially 32 shrews, how many shrews will there be is 21 months?
1. The doubling time of a population of hamsters is 1.2 years. If there are initially 43 hamsters, how many hamsters will there be is 7 years?
1. A certain cancerous tumor doubles in size in three months. If the initial size of the tumor is two cells, how many months will it take for the tumor to grow to 60,000 cells? How many cells will there be in 1.5 years? In three years?
1. A certain cancerous tumor doubles in size in 1.5 months. If the initial size of the tumor is eight cells, how many months will it take for the tumor to grow to 40,000 cells? How many cells will there be in six months? In 2.5 years?
1. A bird population on a certain island has an annual growth rate of 1.5% per year. Approximate the number of years it will take the population to double. If the initial population is 130 birds, use it to find the bird population of the island in 14 years.
1. The beaver population on Kodiak Island has an annual growth rate of 1.2% per year. Approximate the number of years it will take the population to double. If the initial population is 32 beavers, use it to find the population of beavers on the island in 20 years.
1. The black-footed ferret population in Arizona has an annual growth rate of 0.5% per year. Approximate the number of years it will take the population to double. If the initial population is 230 ferrets, use it to find the ferret population in AZ in 12 years.
1. The Mexican gray wolf population in southern Arizona increased from 72 individuals to 92 individuals from 2012 to 2013. What is the annual growth rate? Approximate the number of years it will take the population to double. Create the doubling time model and use it to find the population of Mexican gray wolves in 10 years.
1. There is a small population of Sonoran pronghorn antelope in a captive breeding program on the Cabeza Prieta National Wildlife Refuge in southwest Arizona. Recently, the population increased from 122 individuals to 135 individuals from 2012 to 2013. Approximate the number of years it will take the population to double. Create the doubling time model and use it to find the population of Sonoran pronghorn antelope in 10 years.
1. Lead-209 is a radioactive isotope. It has a half-life of 3.3 hours. Suppose that 56 milligrams of this isotope is created in an experiment, how much is left after 12 hours?
1. Titanium-44 has a half-life of 63 years. If there is 560 grams of this isotope, how much is left after 1200 years?
1. Uranium-232 has a half-life of 68.9 years. If there is 160 grams of this isotope, how much is left after 1000 years?
1. Nickel-63 has a half-life of 100 years. If there is 16 grams of this isotope, how much is left after 145 years?
1. Radium-226 has a half-life of 1600 years. If there is 56 grams of this isotope, how much is left after 145,000 years?
1. The population of wild desert tortoises is decreasing by 72% per year. Approximate the half-life for this population. If there are currently 100,000 tortoises left in the wild, how many will remain in 20 years?
1. The population of pygmy owls is decreasing by 4.4% per year. Approximate the half-life for this population. If there are currently 41 owls left in the wild, how many will remain in five years?
1. Radioactive carbon-14 is used to determine the age of artifacts because it concentrates in organisms only when they are alive. It has a half-life of 5730 years. In 2004, carbonized plant remains were found where human artifacts were unearthed along the Savannah River in Allendale County. Analysis indicated that the plant remains contained 0.2% of their original carbon-14. Estimate the age of the plant remains. (Wikipedia, n.d.)
1. The population of a bird species on an island grows according to the logistic model below. $P(t)=\dfrac{43,240}{1+12 e^{-0.05 t}}\nonumber$Identify the initial population. What will be the bird population in five years? In 150 years? In 500 years? What is the carrying capacity of the bird population on the island?
2. Suppose the population of students at Arizona State University grows according to the logistic model below with the initial population taken from 2009. $P(t)=\dfrac{85,240}{1+0.6 e^{-0.15 t}} \nonumber$Identify the initial population. What will be the campus population in five years? What will be the population in 20 years? What is the carrying capacity of students at ASU?
3. Suppose the population of students at Ohio State University grows according to the logistic model below with the initial population taken from 1970. $P(t)=\dfrac{90,000}{1+4 e^{-0.06 t}} \nonumber$Identify the initial population. What will be the campus population in five years? What will be the population in 20 years? What is the carrying capacity of students at OSU?
4. Suppose that in a certain shrimp farm, the shrimp population is modeled by the logistic model below where t is measured in years. $P(t)=\dfrac{120,000}{1+25 e^{-0.12 t}} \nonumber$Find the initial population. How long will it take for the population to reach 6000 shrimp? What is the carrying capacity of shrimp on the farm?
5. Suppose that in a certain oyster farm, the oyster population is modeled by the logistic model below where t is measured in years. $P(t)=\dfrac{18,000}{1+19 e^{-0.36 t}} \nonumber$Find the initial population. How long will it take for the population to reach 2000 oysters? What is the carrying capacity of oysters on the farm?

This page titled 4.5: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.